Solve for a square root of 5a+2- square root of 7a-8=0

$\sqrt{5 a + 2} - \sqrt{7 a - 8} = 0$

Step 1

Add $\sqrt{7 a - 8}$ to both sides of the equation.

$\sqrt{5 a + 2} = \sqrt{7 a - 8}$

Step 2

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{5 a + 2}}^{2} = {\sqrt{7 a - 8}}^{2}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 a + 2}$ as ${(5 a + 2)}^{\frac{1}{2}}$ .

${({(5 a + 2)}^{\frac{1}{2}})}^{2} = {\sqrt{7 a - 8}}^{2}$

Simplify the left side.

Simplify ${({(5 a + 2)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(5 a + 2)}^{\frac{1}{2}})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(5 a + 2)}^{\frac{1}{2} \cdot 2} = {\sqrt{7 a - 8}}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(5 a + 2)}^{\frac{1}{2} \cdot 2} = {\sqrt{7 a - 8}}^{2}$

Rewrite the expression.

${(5 a + 2)}^{1} = {\sqrt{7 a - 8}}^{2}$

Simplify.

$5 a + 2 = {\sqrt{7 a - 8}}^{2}$

Simplify the right side.

Rewrite ${\sqrt{7 a - 8}}^{2}$ as $7 a - 8$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7 a - 8}$ as ${(7 a - 8)}^{\frac{1}{2}}$ .

$5 a + 2 = {({(7 a - 8)}^{\frac{1}{2}})}^{2}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$5 a + 2 = {(7 a - 8)}^{\frac{1}{2} \cdot 2}$

Combine $\frac{1}{2}$ and $2$ .

$5 a + 2 = {(7 a - 8)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$5 a + 2 = {(7 a - 8)}^{\frac{2}{2}}$

Rewrite the expression.

$5 a + 2 = {(7 a - 8)}^{1}$

Simplify.

$5 a + 2 = 7 a - 8$

Step 4

Solve for $a$ .

Move all terms containing $a$ to the left side of the equation.

Subtract $7 a$ from both sides of the equation.

$5 a + 2 - 7 a = - 8$

Subtract $7 a$ from $5 a$ .

$- 2 a + 2 = - 8$

Move all terms not containing $a$ to the right side of the equation.

Subtract $2$ from both sides of the equation.

$- 2 a = - 8 - 2$

Subtract $2$ from $- 8$ .

$- 2 a = - 10$

Divide each term in $- 2 a = - 10$ by $- 2$ and simplify.

Divide each term in $- 2 a = - 10$ by $- 2$ .

$\frac{- 2 a}{- 2} = \frac{- 10}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 a}{- 2} = \frac{- 10}{- 2}$

Divide $a$ by $1$ .

$a = \frac{- 10}{- 2}$

Simplify the right side.

Divide $- 10$ by $- 2$ .

$a = 5$

Solve for a square root of 5a+2- square root of 7a-8=0

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